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Mirrors > Home > MPE Home > Th. List > mircgrextend | Structured version Visualization version GIF version |
Description: Link congruence over a pair of mirror points. cf tgcgrextend 28263. (Contributed by Thierry Arnoux, 4-Oct-2020.) |
Ref | Expression |
---|---|
mirval.p | β’ π = (BaseβπΊ) |
mirval.d | β’ β = (distβπΊ) |
mirval.i | β’ πΌ = (ItvβπΊ) |
mirval.l | β’ πΏ = (LineGβπΊ) |
mirval.s | β’ π = (pInvGβπΊ) |
mirval.g | β’ (π β πΊ β TarskiG) |
mirtrcgr.e | β’ βΌ = (cgrGβπΊ) |
mirtrcgr.m | β’ π = (πβπ΅) |
mirtrcgr.n | β’ π = (πβπ) |
mirtrcgr.a | β’ (π β π΄ β π) |
mirtrcgr.b | β’ (π β π΅ β π) |
mirtrcgr.x | β’ (π β π β π) |
mirtrcgr.y | β’ (π β π β π) |
mircgrextend.1 | β’ (π β (π΄ β π΅) = (π β π)) |
Ref | Expression |
---|---|
mircgrextend | β’ (π β (π΄ β (πβπ΄)) = (π β (πβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . 2 β’ π = (BaseβπΊ) | |
2 | mirval.d | . 2 β’ β = (distβπΊ) | |
3 | mirval.i | . 2 β’ πΌ = (ItvβπΊ) | |
4 | mirval.g | . 2 β’ (π β πΊ β TarskiG) | |
5 | mirtrcgr.a | . 2 β’ (π β π΄ β π) | |
6 | mirtrcgr.b | . 2 β’ (π β π΅ β π) | |
7 | mirval.l | . . 3 β’ πΏ = (LineGβπΊ) | |
8 | mirval.s | . . 3 β’ π = (pInvGβπΊ) | |
9 | mirtrcgr.m | . . 3 β’ π = (πβπ΅) | |
10 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircl 28439 | . 2 β’ (π β (πβπ΄) β π) |
11 | mirtrcgr.x | . 2 β’ (π β π β π) | |
12 | mirtrcgr.y | . 2 β’ (π β π β π) | |
13 | mirtrcgr.n | . . 3 β’ π = (πβπ) | |
14 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircl 28439 | . 2 β’ (π β (πβπ) β π) |
15 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mirbtwn 28436 | . . 3 β’ (π β π΅ β ((πβπ΄)πΌπ΄)) |
16 | 1, 2, 3, 4, 10, 6, 5, 15 | tgbtwncom 28266 | . 2 β’ (π β π΅ β (π΄πΌ(πβπ΄))) |
17 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mirbtwn 28436 | . . 3 β’ (π β π β ((πβπ)πΌπ)) |
18 | 1, 2, 3, 4, 14, 12, 11, 17 | tgbtwncom 28266 | . 2 β’ (π β π β (ππΌ(πβπ))) |
19 | mircgrextend.1 | . 2 β’ (π β (π΄ β π΅) = (π β π)) | |
20 | 1, 2, 3, 4, 5, 6, 11, 12, 19 | tgcgrcomlr 28258 | . . 3 β’ (π β (π΅ β π΄) = (π β π)) |
21 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircgr 28435 | . . 3 β’ (π β (π΅ β (πβπ΄)) = (π΅ β π΄)) |
22 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircgr 28435 | . . 3 β’ (π β (π β (πβπ)) = (π β π)) |
23 | 20, 21, 22 | 3eqtr4d 2777 | . 2 β’ (π β (π΅ β (πβπ΄)) = (π β (πβπ))) |
24 | 1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23 | tgcgrextend 28263 | 1 β’ (π β (π΄ β (πβπ΄)) = (π β (πβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 βcfv 6542 (class class class)co 7414 Basecbs 17165 distcds 17227 TarskiGcstrkg 28205 Itvcitv 28211 LineGclng 28212 cgrGccgrg 28288 pInvGcmir 28430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-trkgc 28226 df-trkgb 28227 df-trkgcb 28228 df-trkg 28231 df-mir 28431 |
This theorem is referenced by: mirtrcgr 28461 |
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